21-1-2: Special Right Triangles
Objectives:
1. To discover and use
the properties of
45-45-90 and
30-60-90 right
triangles to solve
problems
•
•
•
•
Assignment:
P. 294: 10-12
P. 297-298: 12-19
P. 299-300: 1-14
Challenge Problems
You will be able to use the
properties of 45-45-90 right
triangles to solve problems
Investigation 1
In this investigation, you will discover a
relationship between the lengths of the
legs and the hypotenuse of an isosceles
right triangle.
This triangle is also referred to
as a 45-45-90 right triangle
because each of its acute
angles measures 45°. Folding
a square in half can make one
of these triangles.
Investigation 1
Find the length of the hypotenuse of each isosceles
right triangle. Simplify the square root each time
to reveal a pattern.
Investigation 1
Did you notice something interesting about
the relationship between the length of the
hypotenuse and the length of the legs in
each problem of this investigation?
Special Right Triangle Theorem
45°-45°-90° Triangle
Theorem
In a 45°-45°-90°
triangle, the
hypotenuse is 2
times as long as
each leg.
hypotenuse  leg  2
Example 1
Use deductive
reasoning to verify
the Isosceles Right
Triangle Theorem.
Example 2
A fence around a square garden has a
perimeter of 48 feet. Find the approximate
length of the diagonal of this square
garden.
You will be able to use the properties
of 30-60-90 right triangles to solve
problems
Investigation 2
The second special right triangle is the
30-60-90 right triangle, which is half of an
equilateral triangle.
Let’s start by using a little
deductive reasoning to reveal
a useful relationship in
30-60-90 right triangles.
Investigation 2
Triangle 𝐴𝐵𝐶 is equilateral, and
𝐶𝐷 is an altitude.
1. What are 𝑚∠𝐴 and 𝑚∠𝐵?
2. What are 𝑚∠𝐴𝐷𝐶 and
𝑚∠𝐵𝐷𝐶?
3. What are 𝑚∠𝐴𝐶𝐷 and
𝑚∠𝐵𝐶𝐷?
4. Is Δ𝐴𝐷𝐶 ≅ Δ𝐵𝐷𝐶? Why?
5. Is 𝐴𝐷 = 𝐵𝐷? Why?
Investigation 2
Notice that altitude 𝐶𝐷 divides the equilateral
triangle into two right triangles with acute
angles that measure 30° and 60°. Look at
just one of the 30-60-90 right triangles.
How do AC and AD compare?
Conjecture:
In a 30°-60°-90° right triangle, if the side
opposite the 30° angle has length 𝑥, then
the hypotenuse has length -?-.
Investigation 2
Find the length of the indicated side in each right
triangle by using the conjecture you just made.
Investigation 2
Now use the previous conjecture and the
Pythagorean formula to find the length of each
indicated side.
Investigation 2
You should have notice a pattern in your
answers. Combine your observations with
you latest conjecture and state your next
conjecture.
Special Right Triangle Theorem
30°-60°-90° Triangle
Theorem
In a 30°-60°-90°
triangle, the
hypotenuse is twice
as long as the shorter
hypotenuse  2  shorter leg
leg, and the longer leg
longer leg  shorter leg  3
is 3 times as long as
the shorter leg.
Two Special Right Triangles
Example 3
Find the value of each variable. Write your
answer in simplest radical form.
1.
2.
3.
Example 4
Find the value of each variable. Write your
answer in simplest radical form.
1.
2.
3.
Example 5
What is the area of an equilateral triangle
with a side length of 4 cm?
4 cm
4 cm
4 cm
Example 6: SAT
In the figure, what is the ratio of RW to WS?
T
60
30
R
W
S
21-1-2: Special Right Triangles
Objectives:
1. To discover and use
the properties of
45-45-90 and
30-60-90 right
triangles to solve
problems
•
•
•
•
Assignment:
P. 294: 10-12
P. 297-298: 12-19
P. 299-300: 1-14
Challenge Problems